1. You will have to implement custom forms authentication for this.
refer this: Understanding and Implementing ASP.NET Custom Forms Authentication
2. Whenever user chooses the remember me option you will have to deploy a persistent cookie on his computer. this cookie will contain the authentication ticket. if we look the above article the only thing that will change is:
and set the number of days for which you want the cookie/password to be remembered.
Reading this answer will not give you the solution directly. you will have to read the linked article and make the suggested change to get solution to your problem. Let me know if you need more on this.